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arXiv:1104.1432 · Physical Review B

Z 2 spin liquids in the S = 1 2 Heisenberg model on the kagome lattice: A projective symmetry-group study of Schwinger fermion mean-field states

Yuan-Ming Lu, Ying Ran, Patrick A. Lee

Published Jun 22, 2011• 147 citations
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Abstract

Due to strong geometric frustration and quantum fluctuation, the $S=1/2$ quantum Heisenberg antiferromagnet on the $\text{kagome}$ lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg $S=1/2,\text{kagome}$ lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of ${\mathbb{Z}}_{2}$ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of ${\mathbb{Z}}_{2}$ SLs on the $\text{kagome}$ lattice. Among them we found only one gapped ${\mathbb{Z}}_{2}$ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.

DOI
10.1103/physrevb.83.224413

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Z2 [Z subscript 2] spin liquids in the S=1/2 Heisenberg model on the kagome lattice: A projective symmetry-group study of Schwinger fermion mean-field states

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Due to strong geometric frustration and quantum fluctuation, the S = 1/2 quantum Heisenberg antiferromagnet on the kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S = 1/2, kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2 [Z subscript 2] topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2 [Z subscript 2] SLs on the kagome lattice. Among them we found only one gapped Z2 [Z subscript 2] SL (which we call the Z2[0,π]β [Z subscript 2 [0,pi] Beta] state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this Z2[0,π]β [Z subscript 2 [0,pi] Beta] state to be the numerically discovered SL ground state of the HKLM.

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Recent numerical studies have provided strong evidence for a gapped ${Z}_{2}$ quantum spin liquid in the kagome lattice spin-1/2 Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of symmetry fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons obeying either Bose (Schwinger bosons) or Fermi (Abrikosov fermions) statistics, which are then treated within mean-field theory. A long-standing question has been whether these two approaches are truly distinct, or describe the same phase in complementary ways. Here we show that all $8{Z}_{2}$ spin liquid phases in the Schwinger-boson mean-field (SBMF) construction can also be described in terms of Abrikosov fermions, unifying pairs of theories that seem rather distinct. The key idea is that for ${Z}_{2}$ spin liquid states that admit a SBMF description on the kagome lattice, the symmetry fractionalization of visions is uniquely fixed. Two promising candidate states for the kagome Heisenberg model, Sachdev's ${Q}_{1}={Q}_{2}$ SBMF state and the Lu-Ran-Lee ${Z}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ Abrikosov-fermion state, are found to describe the same symmetric spin liquid phase. We expect these results to aid in a complete specification of the numerically observed spin liquid phase. We also discuss a set of ${Z}_{2}$ spin liquid phases in the fermionic parton approach, where spin rotation and lattice symmetries protect gapless edge states that do not admit a SBMF description.

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The S=1/2 kagome-lattice antiferromagnet is one of interesting frustrated quantum spin systems. The system is supposed to exhibit the quantum spin liquid in the ground state, which was proposed as a possible origin of the high-temprature superconductivity. The spin gap is an important physical quantity to characterize the spin liquid behavior. Whether the S=1/2 kagome-lattice antiferromagnet is gapless or has a finite spin gap, is still an unsolved issue. Because any recently developped numerical calculation methods are not enough to determine it in the thermodynamic limit. Our large-scale numerical diazonalization up to 42-spin clusters and a finite-size scaling analysis indicated that the S=1/2 kagome-lattice antiferromagnet is gapless in the thremodynamic limit[1]. It is consistent with the U(1) Dirac spin liquid theory of the kagome-lattice antiferromagnet[2,3]. On the other hand, some density matrix renormalization group (DMRG) calculations supported the gapped Z2 topological spin liquid theory[4,5]. Our recent numerical diagonalization analysis on the magnetization process of a distorted kagome-lattice antiferromagnet indicated that the perfect kagome-lattice system is just on a quantum critical point[6]. It would be a possible reason why it is difficult to determine whether the perfect kagome-lattice antiferromagnet is gapless or gapped. In the present study, a field derivative analysis is applied for the spin gap issue of the kagome-lattice antiferromanget, using the numerical diagonalization data up to 42-spin clusters. It is consistent with our previous conclusion that it is gapless[7]. [1]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80 (2011) 053704. [2]Y. Ran, M. Hermele, P. A. Lee and X. -G. Wen: Phys. Rev. Lett. 98 (2007) 117205. [3]Y. Iqbal, F. Becca, S. Sorella and D. Poilblanc: Phys. Rev. B 87 (2013) 060405(R). [4]S. Yan, D. A. Huse and S. R. White: Science 332 (2011) 1173. [5]S. Nishimoto, N. Shibata and C. Hotta: Nat. Commun. 4 (2013) 2287. [6]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 83 (2014) 104710. [7]H. Nakano and T. Sakai: in preparation.

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Motivated by recent experiments on the vanadium oxyfluoride material DQVOF, we examine possible spin liquid phases on a breathing kagome lattice of $S=1/2$ spins. By performing a projective symmetry group analysis, we determine the possible phases for both fermionic and bosonic ${\mathbb{Z}}_{2}$ spin liquids on this lattice, and establish the correspondence between the two. The nature of the ground state of the Heisenberg model on the isotropic kagome lattice is a hotly debated topic, with both ${\mathbb{Z}}_{2}$ and U(1) spin liquids argued to be plausible ground states. Using variational Monte Carlo techniques, we show that a gapped ${\mathbb{Z}}_{2}$ spin liquid emerges as the clear ground state in the presence of this breathing anisotropy. Our results suggest that the breathing anisotropy helps to stabilize this spin liquid ground state, which may aid us in understanding the results of experiments and help to direct future numerical studies on these systems.

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Due to strong geometric frustration and quantum fluctuation, the S=1/2S=1/2S=1/2 quantum Heisenberg antiferromagnet on the kagome\text{kagome}kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S=1/2,kagomeS=1/2,\text{kagome}S=1/2,kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2{\mathbb{Z}}_{2}Z2​ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2{\mathbb{Z}}_{2}Z2​ SLs on the kagome\text{kagome}kagome lattice. Among them we found only one gapped Z2{\mathbb{Z}}_{2}Z2​ SL (which we call the Z2[0,\ensuremathπ]\ensuremathβ{\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}Z2​[0,\ensuremathπ]\ensuremathβ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this Z2[0,\ensuremathπ]\ensuremathβ{\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}Z2​[0,\ensuremathπ]\ensuremathβ state to be the numerically discovered SL ground state of the HKLM.
Due to strong geometric frustration and quantum fluctuation, the S = 1/2 quantum Heisenberg antiferromagnet on the kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S = 1/2, kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2 [Z subscript 2] topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2 [Z subscript 2] SLs on the kagome lattice. Among them we found only one gapped Z2 [Z subscript 2] SL (which we call the Z2[0,π]β [Z subscript 2 [0,pi] Beta] state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this Z2[0,π]β [Z subscript 2 [0,pi] Beta] state to be the numerically discovered SL ground state of the HKLM.
Motivated by recent density-matrix renormalization group (DMRG) calculations [Yan, Huse, and White, Science 332, 1173 (2011)], which claimed that the ground state of the nearest-neighbor spin-1/21/21/2 Heisenberg antiferromagnet on the kagome lattice geometry is a fully gapped spin liquid with numerical signatures of Z2{\mathbb{Z}}_{2}Z2​ gauge structure, and a further theoretical work [Lu, Ran, and Lee, Phys. Rev. B 83, 224413 (2011)], which gave a classification of all Schwinger-fermion mean-field fully symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids on the kagome lattice, we have thoroughly studied Gutzwiller-projected fermionic wave functions by using quantum variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. In particular, we investigated the energetics of all Z2{\mathbb{Z}}_{2}Z2​ candidates (gapped and gapless) that lie in the neighborhood of the energetically competitive U(1) gapless spin liquids. By using a state-of-the-art optimization method, we were able to conclusively show that the U(1) Dirac state is remarkably stable with respect to all Z2{\mathbb{Z}}_{2}Z2​ spin liquids in its neighborhood, and in particular for opening a gap toward the so-called ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state, which was conjectured to describe the ground state obtained by the DMRG method. Finally, we also considered the addition of a small second nearest-neighbor exchange coupling of both antiferromagnetic and ferromagnetic type, and obtained similar results, namely, a U(1) Dirac spin-liquid ground state.
Motivated by recent numerical and experimental studies of the spin-1/21/21/2 Heisenberg antiferromagnet on the kagome lattice, we formulate a many-body model for fermionic spinons introduced by us earlier [Phys. Rev. Lett. 103, 187203 (2009)]. The spinons interact with an emergent U(1)U(1)U(1) gauge field and experience strong short-range attraction in the S=0S=0S=0 channel. The ground state of the model is generically a Z2{Z}_{2}Z2​ liquid. We calculate the edge of the two-spinon continuum and compare the theory to the slave-fermion approach to the Heisenberg model.
Recent numerical studies have provided strong evidence for a gapped Z2{Z}_{2}Z2​ quantum spin liquid in the kagome lattice spin-1/2 Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of symmetry fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons obeying either Bose (Schwinger bosons) or Fermi (Abrikosov fermions) statistics, which are then treated within mean-field theory. A long-standing question has been whether these two approaches are truly distinct, or describe the same phase in complementary ways. Here we show that all 8Z28{Z}_{2}8Z2​ spin liquid phases in the Schwinger-boson mean-field (SBMF) construction can also be described in terms of Abrikosov fermions, unifying pairs of theories that seem rather distinct. The key idea is that for Z2{Z}_{2}Z2​ spin liquid states that admit a SBMF description on the kagome lattice, the symmetry fractionalization of visions is uniquely fixed. Two promising candidate states for the kagome Heisenberg model, Sachdev's Q1=Q2{Q}_{1}={Q}_{2}Q1​=Q2​ SBMF state and the Lu-Ran-Lee ${Z}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ Abrikosov-fermion state, are found to describe the same symmetric spin liquid phase. We expect these results to aid in a complete specification of the numerically observed spin liquid phase. We also discuss a set of Z2{Z}_{2}Z2​ spin liquid phases in the fermionic parton approach, where spin rotation and lattice symmetries protect gapless edge states that do not admit a SBMF description.
The S=1/2 kagome-lattice antiferromagnet is one of interesting frustrated quantum spin systems. The system is supposed to exhibit the quantum spin liquid in the ground state, which was proposed as a possible origin of the high-temprature superconductivity. The spin gap is an important physical quantity to characterize the spin liquid behavior. Whether the S=1/2 kagome-lattice antiferromagnet is gapless or has a finite spin gap, is still an unsolved issue. Because any recently developped numerical calculation methods are not enough to determine it in the thermodynamic limit. Our large-scale numerical diazonalization up to 42-spin clusters and a finite-size scaling analysis indicated that the S=1/2 kagome-lattice antiferromagnet is gapless in the thremodynamic limit[1]. It is consistent with the U(1) Dirac spin liquid theory of the kagome-lattice antiferromagnet[2,3]. On the other hand, some density matrix renormalization group (DMRG) calculations supported the gapped Z2 topological spin liquid theory[4,5]. Our recent numerical diagonalization analysis on the magnetization process of a distorted kagome-lattice antiferromagnet indicated that the perfect kagome-lattice system is just on a quantum critical point[6]. It would be a possible reason why it is difficult to determine whether the perfect kagome-lattice antiferromagnet is gapless or gapped. In the present study, a field derivative analysis is applied for the spin gap issue of the kagome-lattice antiferromanget, using the numerical diagonalization data up to 42-spin clusters. It is consistent with our previous conclusion that it is gapless[7]. [1]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80 (2011) 053704. [2]Y. Ran, M. Hermele, P. A. Lee and X. -G. Wen: Phys. Rev. Lett. 98 (2007) 117205. [3]Y. Iqbal, F. Becca, S. Sorella and D. Poilblanc: Phys. Rev. B 87 (2013) 060405(R). [4]S. Yan, D. A. Huse and S. R. White: Science 332 (2011) 1173. [5]S. Nishimoto, N. Shibata and C. Hotta: Nat. Commun. 4 (2013) 2287. [6]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 83 (2014) 104710. [7]H. Nakano and T. Sakai: in preparation.
Motivated by recent experiments on the vanadium oxyfluoride material DQVOF, we examine possible spin liquid phases on a breathing kagome lattice of S=1/2S=1/2S=1/2 spins. By performing a projective symmetry group analysis, we determine the possible phases for both fermionic and bosonic Z2{\mathbb{Z}}_{2}Z2​ spin liquids on this lattice, and establish the correspondence between the two. The nature of the ground state of the Heisenberg model on the isotropic kagome lattice is a hotly debated topic, with both Z2{\mathbb{Z}}_{2}Z2​ and U(1) spin liquids argued to be plausible ground states. Using variational Monte Carlo techniques, we show that a gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquid emerges as the clear ground state in the presence of this breathing anisotropy. Our results suggest that the breathing anisotropy helps to stabilize this spin liquid ground state, which may aid us in understanding the results of experiments and help to direct future numerical studies on these systems.